If now a, b be base and altitude of one, a′, b′ those of another parallelogram, α, β and α′, β′ their numerical values respectively, and A, A′ their areas, then
| A | = | a | · | b | = | α | · | β | = | αβ | . |
| A′ | a′ | b′ | α′ | β′ | α′β′ |
In words: The areas of two parallelograms are to each other as the products of the numerical values of their bases and altitudes.
If especially the second parallelogram is the unit square, i.e. a square on the unit of length, then α′ = β′ = 1, A′ = u², and we have
| A | = αβ or A = αβ·u². |
| A′ |
This gives the theorem: The number of unit squares contained in a parallelogram equals the product of the numerical values of base and altitude, and similarly the number of unit squares contained in a triangle equals half the product of the numerical values of base and altitude.
This is often stated by saying that the area of a parallelogram is equal to the product of the base and the altitude, meaning by this product the product of the numerical values, and not the product as defined above in § 20.
§ 68. Propositions 24 and 26 relate to parallelograms about diagonals, such as are considered in Book I., 43. They are—
Prop. 24. Parallelograms about the diameter of any parallelogram are similar to the whole parallelogram and to one another; and its converse (Prop. 26), If two similar parallelograms have a common angle, and be similarly situated, they are about the same diameter.
Between these is inserted a problem.